parameterized algorithms for bio-molecule folding · 2012-07-20 · parameterized computation for...

Parameterized Computation for Bio-molecule Folding

Liming Cai RNA Informatics Laboratory

University of Georgia

Athens, Georgia, USA

About this presentation

• New algorithmic graph algorithms needed for bio-molecule structure prediction

– Involving graphs of small tree width

– Parameterized computation

– Engineering parameters

– New applications for FPT framework

Outline

• Background

• Optimal subgraph isomorphism from k-trees

• Maximum spanning k-tree

• Additional applications

Background

Single strand DNA Transcribed to non-coding RNA sequence

Folded into 3D structure

N N

N

O

H

H

5’-u-u-c-c-g-a-a-g-c-u-c-a-a-c-g-g-g-a-a-a-u-g-a-g-c-u-3’

P a

P c

5’ 3’

P u a

P

g

P

CYTOSINE

N

N

N

O

H

H

H

N

N

GUANINE

URACIL ADENINE

N N

O

O

H

N

N

N

N

N

H H

Background

• Tertiary structure:

– Less understood non-canonical interactions

– Only a small number of resolved structures

• Secondary structure:

– (Well understood) canonical base pairs

– Scaffolding tertiary structure

– Well studied

Background

Background

• Modeling structure with graphs

To characterize interaction relationships between elements (e.g., residues) on the sequence (i.e., interaction topology)

• Need to model interactions: – Neighboring element connections through

backbone chaining

– Spatial contacts through energy potentials

– Simplifications: pair-wise, non-geometric

Background

• Each topological structure of a molecule is modeled with backbone graph [Song et al, 2006]

– V: vertices for elements (often residues)

– D: directed edges only for backbone connections

– A: non-directed edges for spatial contacts

D forms exactly a Hamiltonian path

Background

• A backbone graph example

Backbone graph for tRNA tertiary structure (after residues are grouped)

Background

• Backbone graphs for bio-molecule structures are of small tree width.

Tree width distribution of backbone graphs of 515 RNA resolved tertiary structures from PDB/NDB

Similar distributions hold for backbone graphs formulated with residues as vertices, and for 6,000+ protein tertiary structures.

Background

• Structure prediction from molecule sequences 1. Template based methods

– number of structures is a small fraction of number of sequences

– cannot predict new structures

Backbone graph modeling

Comparison alignment

Background

• Structure prediction from molecule sequences 1. Template based methods

comparison alignment

preprocessed

subgraph isomorphism

Background

• Structure prediction from molecule sequences 1. Template based methods

subgraph isomorphism

Background

• Structure prediction from molecule sequences 2. ab initio (de novo) methods

– Prediction based on only the given sequence

– Potentially can predict new structures

preprocessed

Extract the most plausible interaction topology

geometric fitting and shape refinement

Background

• Structure prediction from molecule sequences 2. ab initio (de novo) methods

Finding an optimal spanning k-tree as the most plausible Interaction topology

Background

• OSGI:

• MSkT:

Subgraph isomorphism from k-trees

• k-tree, tree width, and tree decomposition are fundamental notions in coping with graph algorithm efficiency:

e.g., Courcelle’s theorem [Courcelle, 1990]:

Monadic second order (MSO)-logic definable problems admit -time algorithms on graphs of tree width ≤ k

Subgraph isomorphism from k-trees

• The theorem also includes: subgraph isomorphism from fixed source H of tree width ≤ k to host G can be solved in time

There have been several lines of research extending this result for subgraph isomorphism.

Subgraph isomorphism from k-trees

• H is fixed, while G is planar:

solvable in linear time [Eppstein , 1999]

• H has bounded degree but not fixed, while G has tree width ≤ k:

solvable in time [Matousek and Thomas, 1992, Arnborg and Proskurowski, 1989]

• H is k-connected and a partial k-tree while G is another partial k-tree:

solvable in time d [Dessmark et al, 1999]

Subgraph isomorphism from k-trees

• Our OSGI problem is to find an optimal subgraph isomorphism from a backbone partial k-tree H to an arbitrary G.

Subgraph isomorphism from k-trees

• A subgraph isomorphism mapping f requires

injection:

structure preserving:

• Using a tree decomposition for H, structure preserving can be checked along with a dynamic programming.

Subgraph isomorphism from k-trees

• With the Hamiltonian path constraint, the injection mapping property

can be checked along with the dynamic programming.

Proof by induction on the backbone distance between u and v.

Subgraph isomorphism from k-trees

• OSGI from backbone partial k-tree can be solved in time f for some small integer c > 0 [Song et al, 2006]

Subgraph isomorphism from k-trees

• Backbone does not reduce the parameterized complexity of the problem.

W[1]-hard, by a reduction from k-clique

Subgraph isomorphism from k-trees

• Additional engineering parameterization on the OSGI, with given candidate sets

and bounded map-width m.

Subgraph isomorphism from k-trees

• OSGI problem, parameterized with k of (k-tree) and map width m, is solvable in time

for some constant c and polynomial p.

by following the result of Song et al, [2005].

Subgraph isomorphism from k-trees

• An alternative approach to the OSGI problem

Is there a way for “non-MSO-definable” problems, such as subgraph isomorphism, to be redefined as (transformed to) MSO-definable sets over graphs of small tree width?

Subgraph isomorphism from k-trees

• An alternative approach to the OSGI problem

E.g., subgraph isomorphism from H to G conveniently corresponds to a size |VH| clique over the product graph H x G , in which

Subgraph isomorphism from k-trees

• Any clique would have to satisfy all conditions set for edges in E.

But for SGI over backbone graphs, all conditions can be checked locally, we only need to focus on induced subgraph by each tree bag.

• Thus, the dense induced product subgraph may be replaced with a graph of smaller tree width.

Subgraph isomorphism from k-trees

• Edge ([1, 4], [2, 8]) represented by the independent set involving 2log |VG| + 1 vertices

Subgraph isomorphism from k-trees

• The transformed product graph has tree width ≤O((k+1)log|VG|) instead of (k+1)|VG|.

Maximum spanning k-trees

• Problem definition (MSkT)

Maximum spanning k-trees

• Finding maximum spanning (partial) k-trees from a given graph,

• K=1, the same as minimum spanning tree

• NP-hard even for k=2 [Bern 1987]

Maximum spanning k-trees

• Remain NP-hard for many classes of restricted graphs [Leizhen Cai and Maffray 1993]

– graphs of degree ≤ 3k + 2

– planar graphs

– split graphs

Background

• What type of graphs allow efficient algorithms for determining spanning k-tree?

– Decision problem on split-comparability graphs [Leizhen Cai and Maffray, 1993]

– Not applicable to the optimization problem MSkT

Maximum spanning k-trees

• MSkT over backbone graphs can be solved in time [Samad and Cai, 2012]

• It is not known if the efficiency can be further improved or it is W[1]-hard.

Maximum spanning k-trees

• The algorithm is dynamic programming, taking advantage of a number properties of MSkT on backbone graph.

• Properties are about child-parent relationships of (k+1)-cliques in a k-tree (in some ordering)

Maximum spanning k-trees

• Main Property:

if

then either all or none of

are in the subtree rooted at a single child (k+1)-clique of .

Maximum spanning k-trees

• Theorem: Any (k+1)-clique in a k-tree has at most (k+2) children.

• For dynamic programming, we use a canonical form of a (k+1)-clique sequence in a k-tree, such that each (k+1)-clique has at most two children.

Maximum spanning k-trees

• For every (k+1)-clique and every importable set , we compute the maximum spanning sub k-tree rooted at of .

Maximum spanning k-trees

• The DP table has dimensions

Each entry requires a factor of time

for checking all x’s and y’s

Each entry needs an additional factor of

for enumerating

Maximum spanning k-trees

• We do not know yet where MSkT stands in the W-hierarchy, though

we suspect that it is at least W[1]-hard because

using O(klog n) amount of nondeterminism to guess does not seem to solve the problem.

Maximum spanning k-trees

• Regardless where MSkT stands in the W-hierarchy, the time complexity is too high;

• Is Maximum spanning k-path (MSkP) easier? e.g., solvable in time or better?

Maximum spanning k-trees

• If so, how about restricting the number of branches in the desired k-tree? (biologically still meaningful)

• Are there other engineering parameters making the computation problems easier?

Applications of MSkT solvers

1. Bio-molecule folding

– ab initio structure prediction from single sequence

• A complete graph can be formulated from a given molecule sequence, with edge weights for potentials of interactions between residues

• Parameter value for k is chosen

• MSkT answer gives a most plausible topological graph based on interaction potentials

Applications of MSkT solvers

1. Bio-molecule folding

Note: the result of MSkT is not geometrical structure.

• Incorporating geometric constraints into interaction potentials (non-pairwise, however)

• From topology to geometry

Applications of MSkT solvers

1. Bio-molecule folding

Geometric modeling

– k=2, modeling (k+1)-cliques with triangles

– suitable for secondary structure prediction

Applications of MSkT solvers 1. Bio-molecule folding (k=2, n=10)

1 10

2 9

3 8

4 7

5 6

1, 2, 10 2, 9, 10 2, 3, 9 3, 8, 9 3, 4, 8 4, 7, 8 4, 5, 7 5, 6, 7

RNA stem-loop (not known)

A path decomposition corresponding to the 2-path

1 10 2 9 3 8 4 7 5 6

A predicted (partial) 2-path for k=2, n=10

Applications of MSkT solvers 1. Bio-molecule folding (k=2, n=10)

1 10

2 9

3 8

4 7

5 6

1, 2, 10 2, 9, 10 2, 3, 9 3, 8, 9 3, 4, 8 4, 7, 8 4, 5, 7 5, 6, 7

RNA stem-loop (not known)

A path decomposition corresponding to the predicted 2-path

1 2

10 9

3

8

4 7

5

6

1 2

10 9

8

3 4

7

5

6

Triangle model before scaling

Triangle model after scaling

Applications of MSkT solvers

1. Bio-molecule folding

Geometric modeling

– k=3, modeling (k+1)-cliques with tetrahedrons

– can capture most tertiary structures

Applications of MSkT solvers

1. Bio-molecule folding (k=3, n=9)

1

2 6

3 5

4

7

9

8

1 2 9 3 8 4 7 5 6

RNA pseudoknot (not known)

A predicted (partial) 3-path for k=3, n=9

1, 2, 3, 4

1, 2, 4, 7

2, 4, 6, 7

4, 5, 6, 7

4, 5, 7, 9

5, 7, 8, 9

A path decomposition corresponding to the predicted 3-path

Applications of MSkT solvers

1. Bio-molecule folding (k=3, n=9)

1

2 6

3 5

4

7

9

8

RNA pseudoknot (not known)

1, 2, 3, 4

1, 2, 4, 7

2, 4, 6, 7

4, 5, 6, 7

4, 5, 7, 9

5, 7, 8, 9

A path decomposition corresponding to the predicted 3-path

Tetrahedron modeling Without scaling

backbone

interaction

Applications of MSkT solvers

2. Formal language theory

Languages recognized

Parsing process Grammar rules

CYK algorithm Context-free Tree Context-free rules

Our algorithm Mildly context-sensitive

k-tree Mildly context-sensitive rules?

applications molecule sequences

Molecule structures

a a c g u u c c c c u c u a g a c c

S

S

S

S

S aSu L aL

S uSa L cL

S gSc L a

S cSg L c

S L

• Each derivation tree corresponds to a structure.

Applications of MSkT solvers • Stochastic Context-free Grammars (SCFGs)

L

L

L

L

[Lari and Young’90, Sakakibara et al’94]

S aSu

S cSg

S gSc

S uSa

S a

S c

S g

S u

S SS

1. A CFG

S aSu

acSgu

accSggu

accuSaggu

accuSSaggu

accugScSaggu

accuggSccSaggu

accuggaccSaggu

accuggacccSgaggu

accuggacccuSagaggu

accuggacccuuagaggu

2. A derivation of “accuggacccuuagaggu” 3. Corresponding structure

Applications of MSkT solvers

Applications of MSkT solvers

3. DNA nanotechnology

Using DNA base pairs as basic construct to build complex with precisely controlled nanoscale features.

Applications of MSkT solvers

3. DNA nanotechnology

Can efficient algorithms for MSkT problem (and/or alike) serve critical roles for investigating DNA nanotechnology ?

Conclusion

• Introduced two types of parameterized computation problems on backbone graphs involving k-trees,

• Motivated by bio-molecule folding,

• Additional applications in formal language theory and DNA nanotechnology,

• Open problems, in particular, further engineering parameters for efficiency improvement.

Acknowledgement

• Abdul Samad Pooya Shareghi

• Xiuzhen Huang Russell Malmberg

• NSF NIH